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Generalized perfect arrays and Menon difference sets. (English) Zbl 0767.05030
Given an $$s_ 1\times\cdots\times s_ r$$ integer-valued array $$A$$ and a $$(0,1)$$ vector $${\mathbf z}=(z_ 1,\dots,z_ r)$$, form the array $$A'$$ from $$A$$ by recursively adjoining a negative copy of the current array for each dimension $$i$$ where $$z_ i=1$$. $$A$$ is a generalized perfect array type $${\mathbf z}$$ if all periodic autocorrelation coefficients of $$A'$$ are zero, except for shifts $$(u_ 1,\dots,u_ r)$$ where $$u_ i\equiv 0\pmod{s_ i}$$ for all $$i$$. The array is perfect if $${\mathbf z}=(0,\dots,0)$$ and binary if the array elements are all $$\pm 1$$. A non- trivial perfect binary array (PBA) is equivalent to a Menon difference set in an Abelian group.
Using only elementary techniques, we prove various construction theorems for generalized perfect arrays and establish conditions on their existence. We show that a generalized PBA whose type is not $$(0,\dots,0)$$ is equivalent to a relative difference set in an Abelian factor group. We recursively construct several infinite families of generalized PBAs, and deduce nonexistence results for generalized PBAs whose type is not $$(0,\dots,0)$$ from well-known nonexistence results for PBAs. A central result is that a PBA with $$2^{2y} 3^{2u}$$ elements and no dimension divisible by 9 exists if and only if no dimension is divisible by $$2^{y+2}$$. The results presented here include and enlarge the set of sizes of all previously known generalized PBAs.
Reviewer: J.Jedwab (Bristol)

##### MSC:
 05B10 Combinatorial aspects of difference sets (number-theoretic, group-theoretic, etc.) 05B15 Orthogonal arrays, Latin squares, Room squares
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